❖ In this lesson, we are solving Two Linear Systems \(Ax=b\) and \(Ax=c\) using the inverse.

❖ To find the inverse of a Matrix, we have mentioned two ways to answer. Then, we used the inverse of a Matrix to solve Two Linear Systems.

❖ Question \(1\), we have solved it in two methods:

The first method:

You need to write an Augmented Matrix containing the original Matrix and the Identity Matrix,

Then,

You need to convert the original Matrix into the Identity Matrix using Elementary Row Operations

(Apply Reduced Row Echelon Form (RREF) for the left side of the Augmented Matrix).

The Identity Matrix will convert into the inverse of the original Matrix as long as you apply the same Elementary Row Operations for the Augmented Matrix.

The second method:

You need to do a simple formula to find the inverse of a Matrix.

❖ Question \(2\), we have used the inverse of a Matrix to solve two Linear Systems Ax=b and Ax=c.

Simply, we do:

\(x=A^{-1}\) b for linear system \(Ax=b\),

and

\(x=A^{-1} c\) for linear system \(Ax=c\).

To illustrate how we do this

Since \(A^{-1}\) exists, then, we can multiply both sides of the Linear Systems \(Ax=b\) and \(Ax=c\) by \(A^{-1}\) from the left.

Then we have

\(A^{-1} Ax = A^{-1} b\) for linear system \(Ax=b\),

and

\(A^{-1} Ax = A^{-1} c\) for linear system \(Ax=c\).

We can simplify this to

\(I_2 x = A^{-1} b\) for linear system \(Ax=b\),

and

\(I_2 x = A^{-1} c\) for linear system \(Ax=c\).

(we use \(A A^{-1} = I_2\))

We can simplify this to

\(x = A^{-1} b\) for linear system \(Ax=b\),

and

\(x = A^{-1} c\) for linear system \(Ax=c\).

❖ Also, we have explained in this lesson that if the inverse of the Matrix Does Not Exist (DNE).

Note:

If \(A\) inverse exists, then \(A A^{-1} = A^{-1} A =\) Identity Matrix

If \(A\) is not a Square Matrix, then the inverse of Matrix \(A\) is DNE.

The Square Matrix is a Matrix with

the number of Rows \(=\) the number of Columns.

❖ The number of Rows and Columns that a Matrix has is called its Size, Order, or Dimension.